Shrodinger's Cat Paradox: A Possible Solution

I am reading "Beyond Einstein" by Kaku and one quote in the book specially caught my attention.

... the superstring theory provides ... comprehensive way of looking at Schrodinder's cat. Usually, in quantum mechanics, physicists write the Schrodinger wave function of a certain particle. However, the complete quantum mechanical description of the superstring theory requires that we write the Schrodinger wave function of the entire universe... This does not resolve all the philosophical problems associated with Schrodinger's cat; it merely means that the original formulation of the problem ... may be incomplete.

Can't we give the same argument just by sticking to our present quantum theory? Why do we need to bring in superstring theory?

Just as Stat mech guy said, to get grant money. Superstring theory is a mathematical curiousity, it does not fulfil any of the requirements for a physical theory eg falsification. However I am betting they get funding from the usual physics sources.

I can't see any connection between schrodingers cat and string theory.

The aspect that the cat is neither dead nor alive until any measurement is made because the state of the uranium atom is "undefined" until any measurement is made.

This is no paradox if you remmember that the "cat" in the sentence above means
an ensamble of identically prepared cats. Some of them will turn dead and the rest
-- alive.
The interesting part is if you can prove that in a single run of the cat experiment, i.e. with
one cat, the outcome is either dead or alive. Never a superposition: dead +/- alive.
This proof exists. If you are interested I can dwell on it.

Never a superposition: dead +/- alive.
This proof exists. If you are interested I can dwell on it.

What stops you, in principle, from considering quantum cat interference experiments ?
Suppose - ok, this is jokingly of course - that the cat is standing on two bars before the experiment. When the cat is "dead" she'll fall in between the bars, and when the cat is live, she'll jump through a hole in a wall in front of her.

Now, imagine we have some "cat-reflector" on the floor of the box, which "shines" the (dead) cat also "through the hole" where she is supposed to be jumping through when she's alive. This will then create an interference pattern between the "dead cat beam" (onto the floor, and the cat reflector, through the hole) and the "live cat beam" (directly through the hole), and so there should be maxima for the |dead>+|live> state which correspond to minima for the |dead>-|live> state and vice versa.
So looking at the position of our cat after the hole, we would measure, not a "live" cat, or a "dead" cat, but rather, a live+dead cat or a live-dead cat.

What stops you, in principle, from considering quantum cat interference experiments ?
Suppose - ok, this is jokingly of course - that the cat is standing on two bars before the experiment. When the cat is "dead" she'll fall in between the bars, and when the cat is live, she'll jump through a hole in a wall in front of her.

Now, imagine we have some "cat-reflector" on the floor of the box, which "shines" the (dead) cat also "through the hole" where she is supposed to be jumping through when she's alive. This will then create an interference pattern between the "dead cat beam" (onto the floor, and the cat reflector, through the hole) and the "live cat beam" (directly through the hole), and so there should be maxima for the |dead>+|live> state which correspond to minima for the |dead>-|live> state and vice versa.
So looking at the position of our cat after the hole, we would measure, not a "live" cat, or a "dead" cat, but rather, a live+dead cat or a live-dead cat.

O.K. Suppose the cat is in a alive+dead state. To simplify problem lets assume
for a moment that cat consists of N identical bosons. N is large.
The state alive+dead we can write as |N,0> + |0,N>, namely as the bosonic Schrodinger
cat state, where first slot corresponds to a single particle wave function w0(x) and the second slot to w1(x).
For all we know, to generate an outcome of a single measurement on the state we
draw one point (positions of all N particles) from 3N dimensional probability density
given by the N-body wave function modulus squared.

It turns out that this probability density for large N has two SEPARATE nonzero sectors:
one for |N,0> and one for |0,N>. I you draw just one point out of it, the point will
belong to one of the sectors. The cat turns to either alive or dead in the single run.

Similar in spirit argument works for fermions and distiguishible particles. You don't
have to belive my words. You can check the calculations yourself. They are available
through arXives.

This is no paradox if you remmember that the "cat" in the sentence above means
an ensamble of identically prepared cats. Some of them will turn dead and the rest
-- alive.
The interesting part is if you can prove that in a single run of the cat experiment, i.e. with
one cat, the outcome is either dead or alive. Never a superposition: dead +/- alive.
This proof exists. If you are interested I can dwell on it.

Cheers!

But again (as in previous discussion on this issue), you're ignoring the fact that there ARE consequences of the dead and alive superposition. The coherence gap measured in Delft/Stony Brook experiments clearly is one such example. The situation of either dead or alive will NOT produce such consequences.

Just as Stat mech guy said, to get grant money. Superstring theory is a mathematical curiousity, it does not fulfil any of the requirements for a physical theory eg falsification. However I am betting they get funding from the usual physics sources.

I can't see any connection between schrodingers cat and string theory.

What are the usual suspects? And why are they so keen to fund something that is non falsifiable, and thus could ultimately be of no practical use? Are they banking on the experimental proof turning up or is there some other motivation, what do they know that we don't? Or are some institutions being lead up the garden path?

But again (as in previous discussion on this issue), you're ignoring the fact that there ARE consequences of the dead and alive superposition. The coherence gap measured in Delft/Stony Brook experiments clearly is one such example. The situation of either dead or alive will NOT produce such consequences.

Zz.

Of course there are consequences! The Shroedinger cat state is fully coherent (i.e. pure
, as oposite to mixed) with its all ability to interfere and so on. As any other quantum state that is pure. More, this state can never be classical, no matter how macroscopic
it is. Even if you know the state exactly you can only guess what a measurement will reveal in a single run. Whereas with classical object when you know its "classical state"
(positions + velocities), you know exactly the measurement outcome.

However, when you sample the corresponding probability density for a Schrodinger cat state to generate a single measurement outcome, you get only cat dead or alive. Nothing in between.

However, when you sample the corresponding probability density for a Schrodinger cat state to generate a single measurement outcome, you get only cat dead or alive. Nothing in between.

No. The point is (as Zapper on a more serious note said), if the pure state with the quantum cat, the cat mirror and the hole is |live> + |dead> cat, then, if you repeat this experiment a 1000 times, you will find your cat only in certain positions, and never in others (interference fringes in cat position).

While if you only had a probability density "live" (50%) and "dead" (50%), you would find your cat uniformly distributed in position: no interference fringes in cat position.

Note: of course there's a problem with entanglement with internal degrees of freedom, for a genuinly dead or live cat (that is, its internal state is different, and entangled with its "trajectory"). So it would be necessary to insert a "cat ressurection device" in the falling cat beam, or a cat killing device in the direct beam, to erase the difference in internal states, so that they can factor out and allow for a pure position interference), so that the livelyness of the cat, after "interference" is not an indication of which path information.

Of course there are consequences! The Shroedinger cat state is fully coherent (i.e. pure
, as oposite to mixed) with its all ability to interfere and so on. As any other quantum state that is pure. More, this state can never be classical, no matter how macroscopic
it is. Even if you know the state exactly you can only guess what a measurement will reveal in a single run. Whereas with classical object when you know its "classical state"
(positions + velocities), you know exactly the measurement outcome.

However, when you sample the corresponding probability density for a Schrodinger cat state to generate a single measurement outcome, you get only cat dead or alive. Nothing in between.

Cheers!

I don't believe so. In fact, based on what understood from the formulation based on Tony Leggett's paper[1], there WILL be a difference between (i) superposition of states of the single cat, measured repeatedly versus (ii) a many dead cats and many alive cats, measured one at a time many times. The coherence energy gap measured in those SQUID experiments can be obtained from a single measurement, in principle, and will only occur if there is a superposition of the direction of the supercurrent current flow. In a superconducting state, the whole supercurrent is a single entity, i.e. it is not "many cats".

No. The point is (as Zapper on a more serious note said), if the pure state with the quantum cat, the cat mirror and the hole is |live> + |dead> cat, then, if you repeat this experiment a 1000 times, you will find your cat only in certain positions, and never in others (interference fringes in cat position).

While if you only had a probability density "live" (50%) and "dead" (50%), you would find your cat uniformly distributed in position: no interference fringes in cat position.

Note: of course there's a problem with entanglement with internal degrees of freedom, for a genuinly dead or live cat (that is, its internal state is different, and entangled with its "trajectory"). So it would be necessary to insert a "cat ressurection device" in the falling cat beam, or a cat killing device in the direct beam, to erase the difference in internal states, so that they can factor out and allow for a pure position interference), so that the livelyness of the cat, after "interference" is not an indication of which path information.

Could you, please, write the state that undergoes the measurement? For simplicity, suppouse that we start with a bosonic Schrodinger cat state: |N,0> + |0,N>,
if it is not a problem. What happens with it after the mirror, the hole ,etc? The final state,
please.

I don't believe so. In fact, based on what understood from the formulation based on Tony Leggett's paper[1], there WILL be a difference between (i) superposition of states of the single cat, measured repeatedly versus (ii) a many dead cats and many alive cats, measured one at a time many times. The coherence energy gap measured in those SQUID experiments can be obtained from a single measurement, in principle, and will only occur if there is a superposition of the direction of the supercurrent current flow. In a superconducting state, the whole supercurrent is a single entity, i.e. it is not "many cats".

Zz.

[1] A.J. Leggett, J. Phys. Condens. Matt., v.14, p.415 (2002).

You are right, but in this case I did not find such superposition to be strange : as I seem to remember from the paper (some time ago) it was about two supercurrents going in opposite directions through a tube connected by a Josephson bridge. The whole difficulty with superposition manifests itself when measuring superpositions of product states. In the latter paper, ``measurement'' was not causing a reduction of the state (a point the authors should have payed more attention to from the QM perspective, albeit it is intuitively clear).

I don't believe so. In fact, based on what understood from the formulation based on Tony Leggett's paper[1], there WILL be a difference between (i) superposition of states of the single cat, measured repeatedly versus (ii) a many dead cats and many alive cats, measured one at a time many times. The coherence energy gap measured in those SQUID experiments can be obtained from a single measurement, in principle, and will only occur if there is a superposition of the direction of the supercurrent current flow. In a superconducting state, the whole supercurrent is a single entity, i.e. it is not "many cats".

I agree on your points (i) and (ii). There is difference! I didn't considered (ii) so far
on this forum. I was only concerned with (i)-- a pure Schrodinger cat state. Still the
mathematical structure of the corresponding probability density is such that in a single
measurement you will get almost certainly cat dead or cat alive. The more particles
involved the greater the certainty.

A superconducting state represents many identically prepared superconductors.
Somtimes however (BCS superconductor, BEC, ...) a single quantum object
consisting of many particles gives the same averages (when averaged over all particles)
as an ensemble of single quantum objects. A member of an ensemble has the same
properties as the ensemble itself.

In these cases you can really apply QM to a single quantum object! But this is only in the
large N limit and for very special quantum states.

You are right, but in this case I did not find such superposition to be strange : as I seem to remember from the paper (some time ago) it was about two supercurrents going in opposite directions through a tube connected by a Josephson bridge. The whole difficulty with superposition manifests itself when measuring superpositions of product states. In the latter paper, ``measurement'' was not causing a reduction of the state (a point the authors should have payed more attention to from the QM perspective, albeit it is intuitively clear).

But having two currents going in opposite direction isn't the case here. If it were, then there's nothing unusual about this experiment and the results would never get into a journal like Nature.

If we buy the QM description of superconductivity, then the supercurrent is a single, coherent entity. It is this single entity that has two different directions of transport. The state description isn't composed of a percentage of the supercurrent having one direction, while the rest goes the other way. The description is one in which the supercurrent has both directions, and the ratio of the probability of one versus the other depends on the external magnetic field. The predicted outcome of the same measurement will not produce the same results if you simply have two independent currents going in opposite directions.

I agree on your points (i) and (ii). There is difference! I didn't considered (ii) so far
on this forum. I was only concerned with (i)-- a pure Schrodinger cat state. Still the
mathematical structure of the corresponding probability density is such that in a single
measurement you will get almost certainly cat dead or cat alive. The more particles
involved the greater the certainty.

I don't think there's any argument here in terms of the outcome of a measurement that measures one or the other. However, I'm arguing about the fact that the superposition does create a series of testable consequences. If by opening the box you make a measurement of the dead or alive characteristics of the cat, then don't open the box and measure an operator that is non-commuting to "opening the box". This would not cause the state to choose one over the other and thus, the superposition is maintained. It is this measurement that I'm focusing on, and it is this measurement that has been done in those two experiments that I cited. They didn't measure if the supercurrent is moving one way or the other because the very act of such a measurement will give then only one answer. Instead, they measure the consequence of such a superposition by measuring the energy spectrum instead. The existence of the coherent energy gap is consistent with the superposition of states and not simply a matter of having a current going one way or the other.

The very same principle also explains the bonding-antibonding bands we have in solid state physics and chemistry. An electron residing in only one of the bands would not cause an interference with itself from the other band to cause the presence of bonding and antibonding states. It must be in a superposition of both bands for that to occur.